User:Nippashish/sandbox

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Distribution	$h(x)$	$T(x)$	$\eta$	$A(\eta )$
Bernoulli	$1$	$x$	$\log \left({\frac {p}{1-p}}\right)$	$\log(1+\exp(\eta ))$
Binomial	${n \choose x}$	$x$	$\log \left({\frac {p}{1-p}}\right)$	$\log(1+\exp(\eta ))$
Poisson	${\frac {1}{x!}}$	$x$	$\log(\lambda )$	$\exp(\eta )$
Normal (known variance)	${\frac {1}{\sqrt {2\pi }}}\exp \left(-{\frac {x^{2}}{2\sigma ^{2}}}\right)$	${\frac {x}{\sigma }}$	${\frac {\mu }{\sigma }}$	${\frac {\mu ^{2}}{2\sigma ^{2}}}$
Normal	${\frac {1}{\sqrt {2\pi }}}$	${\begin{bmatrix}x\\x^{2}\end{bmatrix}}$	${\begin{bmatrix}{\frac {\mu }{\sigma ^{2}}}\\-{\frac {1}{2\sigma ^{2}}}\end{bmatrix}}$	${\frac {\mu ^{2}}{2\sigma ^{2}}}+\log |\sigma |$
Exponential	$1$	$x$	$-\lambda$	$-\log(\lambda )$
Gamma (rate)	$1$	${\begin{bmatrix}x\\\log(x)\end{bmatrix}}$	${\begin{bmatrix}-{\frac {1}{\theta }}\\k-1\end{bmatrix}}$	$\log(\Gamma (k))+k\log(\theta )$
Chi-squared	$\exp \left(-{\frac {x}{2}}\right)$	${\frac {k}{2}}-1$	$\log(x)$	${\frac {k}{2}}\log(2)+\log(\Gamma ({\frac {k}{2}}))$